Energy resolution

The hypernuclear mass resolution is governed by the following items in the present experiment. Contribution of the each item to the energy resolution are summarized in Table 2. The proposed spectrometer system is expected to achieve the energy resolution of 300-400 keV (FWHM), depending on the choice of targets.

1. HKS momentum resolution
   With a Monte Carlo simulation, the momentum resolution of the HKS was estimated to be 80 keV/$c$ (rms) for a spatial resolution of 200$\mu$m of the chambers. The detail discussion is given in section 3.4.

2. Beam momentum resolution
   Beam momentum resolution is required to be 1 $\times 10^{-4}$. It corresponds to 180 keV/$c$ (FWHM) for the 1.8 GeV/$c$ electron beam. It is necessary to monitor long term fluctuation of the beam central momentum, which will be corrected in the off-line analysis, if it is significant.

3. Enge momentum resolution
   Assuming a chamber spatial resolution of 200$\mu$m (rms) and an angular resolution of 2 mr (rms), the momentum resolution of the Enge spectrometer is estimated to be 120 keV (FWHM). (see section 6.1).
   

4. Kinematical broadening due to uncertainty of the K$^+$ scattering angle
   The uncertainty of the K$^+$ emission angle is dominated by the multiple scattering through the materials between the target and the chambers (uncertainty due to the spatial resolution of the chambers are less than 0.2 mrad). Contribution of the following materials are taken into account: the target (100 mg/cm$^2$), vacuum windows (kevlar 0.008'', mylar 0.005''), helium bag (95 cm) and chambers (mylar 0.0045'', argon 5.08 cm @ STP).

   Total uncertainty of the K$^+$ angle was estimated to be 3.3 mrad (rms) for the carbon target. This angular uncertainty corresponds to 152 keV (FWHM) ambiguity to the $^{12}_\Lambda$B mass. Similarly, this effect to $^{28}_\Lambda$Al (Si target) is 64 keV (FWHM).

5. Momentum loss in the target
   The momentum loss in the target was calculated for 1.2 GeV/$c$ K$^+$ assuming Vavilov distribution, which directly contributes to the mass resolution. The average momentum loss of 1.2 GeV/$c$ K$^+$ in the carbon 100 mg/cm$^2$ was 195 keV/$c$, corresponding to the energy resolution of 180 keV (FWHM).

表 2: The energy resolution of the HKS system

Item	Contribution to the resolution (keV, FWHM)
Target	C	Si	V	Y
HKS momentum	170
Beam momentum	$\le$ 180
Enge momentum	120
K$^+$ angle	152	64	36	20
Target (100 mg/cm$^2$)	$\le$ 180	$\le$ 171	$\le$ 148	$\le$ 138
Overall	$\le$ 360	$\le$ 330	$\le$ 315	$\le$ 310